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\begin{document}

\begin{center}
	\LARGE{Math 31 - Homework 5}
	
	\vspace{2mm}
	\large{Due Friday, August 3}
\end{center}

\subsection*{Easy}
\paragraph{1.} Let $G = \la a \ra$ be a cyclic group of order $n$. You showed on a previous homework that if $m \mid n$, then $G$ has a unique subgroup of order $m$, namely
\[
	N = \la a^{n/m} \ra.
\]
Moreover, $N \normal G$ since $G$ is abelian.
\begin{thmenum}
	\item How many cosets of $N$ are there in $G$? Find them.
	\item Given two cosets of $N$, what is their product? Verify that each coset is a power of the coset $Na$.
	\item Conclude that $G/N$ is a cyclic group. What is its order?
\end{thmenum}

\paragraph{2.} [Herstein, Section 2.6 \#2] Recall that $\R^\times$ is the group of nonzero real numbers (under multiplication), and let $N = \{-1, 1\}$. Show that $N$ is a normal subgroup of $\R^\times$, and that $\R^\times/N$ is isomorphic to the group of positive real numbers under multiplication. [\textbf{Hint:} Use the First Homomorphism Theorem.]

\paragraph{3.} [Herstein, Section 3.2 \#2] Find the cycle decomposition and order of each of the following permutations.
\begin{thmenum}
	\item $\left( \begin{array}{ccccccccc}
			1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\
			3 & 1 & 4 & 2 & 7 & 6 & 9 & 8 & 5
		\end{array} \right)$
		
	\item $\left( \begin{array}{ccccccc}
			1 & 2 & 3 & 4 & 5 & 6 & 7  \\
			7 & 6 & 5 & 4 & 3 & 2 & 1
		\end{array} \right)$
		
	\item $\left( \begin{array}{ccccccc}
			1 & 2 & 3 & 4 & 5 & 6 & 7  \\
			7 & 6 & 5 & 3 & 4 & 2 & 1
		\end{array} \right) \left( \begin{array}{ccccccc}
			1 & 2 & 3 & 4 & 5 & 6 & 7  \\
			2 & 3 & 1 & 5 & 6 & 7 & 4
		\end{array} \right)$
\end{thmenum}

\paragraph{4.} [Herstein, Section 3.3 \#1] Determine whether each permutation is even or odd.
\begin{thmenum}
	\item $\left( \begin{array}{ccccccccc}
			1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\
			2 & 4 & 5 & 1 & 3 & 7 & 8 & 9 & 6
		\end{array} \right)$
		
	\item $(1 \quad 2 \quad 3 \quad 4 \quad 5 \quad 6)(7 \quad 8 \quad 9)$
	
	\item $(1 \quad 2 \quad 3 \quad 4 \quad 5 \quad 6)(1 \quad 2 \quad 3 \quad 4 \quad 5 \quad 7)$
	
	\item $(1 \quad 2)(1 \quad 2 \quad 3)(4 \quad 5)(5 \quad 6 \quad 8)(1 \quad 7 \quad 9)$
\end{thmenum}

\paragraph{5.} [Herstein, Section 3.3 \#5] Suppose you are told that the permutation
\[
	\left( \begin{array}{ccccccccc}
			1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\
			3 & 1 & 2 &  &  & 7 & 8 & 9 & 6
		\end{array} \right)
\]
in $S_9$, where the images of $4$ and $5$ have been lost, is an even permutation. What must the images of 4 and 5 be?

\subsection*{Medium}
\paragraph{6.} Prove that if $G$ is abelian and $N \leq G$, then $G/N$ is abelian. [\textbf{Hint:} You may want to use a result from the last homework assignment.]

\paragraph{7.} If $G$ is a group and $M \normal G$, $N \normal G$, prove that $M \cap N \normal G$. [You proved on an earlier assignment that $M \cap N$ is a subgroup of $G$, so you only need to prove that it is normal.]

\paragraph{8.} Let $G$ be a group. Recall from a previous homework that the \textbf{center} of $G$ is the set $Z(G)$ defined by
	\[
		Z(G) = \left\{ x \in G : xa = ax \text{ for all } a \in G \right\}.
	\]
	You proved that $Z(G)$ is a subgroup of $G$.
	\begin{thmenum}
		\item Prove that $Z(G) \normal G$.
		\item If $G/Z(G)$ is cyclic, prove that $G$ is abelian.
	\end{thmenum}
	
\paragraph{9.} Let $G$ be a group and $H \leq G$. If $[G:H]=2$, prove that $H$ is normal in $G$.

\subsection*{Hard}
\paragraph{10.} Let $G$ be a group, and recall that $\Aut(G)$ is the set of all automorphisms of $G$. You proved on the last homework that $\Aut(G)$ forms a group under composition.
\begin{thmenum}
	\item Given $g \in G$, define a function $\theta_g : G \to G$ by
		\[
			\theta_g(a) = gag^{-1}
		\]
		for all $a \in G$. Show that $\theta_g \in \Aut(G)$. (Such an automorphism is called an \textbf{inner automorphism}.)
		
	\item Let $\Inn(G)$ denote the set of all inner automorphisms of $G$. Then $\Inn(G) \subset \Aut(G)$ by part (a). Show that $\Inn(G)$ is actually a 			subgroup of $\Aut(G)$.
	
	\item Prove that $\Inn(G)$ is a normal subgroup.
\end{thmenum}

















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